7-fold

I have this quote, by Max Bill in his introduction to the catalogue of the Zürich Exhibition in 1947: "the goal of concrete art is to develop objects for mental use, just like people created objects for material use." If anyone has access to the entire document, please let me know. I'd love to read the whole thing.

These large 7-fold or 14-fold patterns with the girih-like polygon set are difficult to generate without gaps.

14-fold Rotational Symmetry, Spiral

This spiral tiling with 14-fold symmetry was built from three equilateral polygons, including a heptagon. There are several ways to fill the center so that the tiling is a complete tessellation. A rhombus and the two polygons other than the heptagon can be used, though the complete tiling becomes 7-fold.

Edge-to-edge heptagons can be followed from the center outward. The tiling might extend infinitely(?)

Girih Process Art

A potentially infinite process controls the development of these images. The process is difficult but not impossible to describe. It could be extremely boring to list all the steps required, and insufficient to help visualize the final image. Just covering a few guidelines, imagine six equilateral polygons including a heptagon and tetradecagon, all constructed with internal angles divisible by π/7. Arrange them edge-to-edge with no gaps and no overlaps. The radial pattern should have seven (or fourteen) fold symmetry, and be extendable forever. In my solution, outer rows of regular heptagons alternate with rows of equilateral hexagons and a rhombus.

As a Design System

Girih Extended, Arabesque, and Seven are design systems similar to girih tiles, a medieval Islamic patterning technique. Like the original girih tiles, they facilitate the generation of complex patterns. Instead of constructing patterns line by line, we can design by tiling polygons. The systems are fast and accurate. The design below was generated in a few hours. The bottom diagram reflects the polygons that are selected from a menu and positioned edge-to-edge with another. The top and middle diagrams are generated automatically from the polygons.

These designs can be scaled from small to architectural applications. The final vector files are suitable for digital prints and processes.

Girih Extended, Arabesque, and Seven designs are reminiscent of Islamic art. The application of scaling, arabesque, and heptagon-based polygons to girih tiles shares much in common with, but extends girih pattern possibilities.

Girih Seven Examples

I’ve published a group of Girih Seven images. The final girih seven polygon set includes a heptagon, tetradecagon, elongated hexagon, rhombus, and bow tie. This follows the original girih tile example. I’ve added a second rhombus, another elongated hexagon, and several scaling tiles, including a trapezoid and kites. My bow tie has eight sides, instead of six, though two trapezoids make one six-sided bow tie. This larger tile set makes it possible to design without gaps, but it’s more difficult than using the pentagon based girih tiles. I haven’t made asymmetric patterns like those that are possible with girih extended. Otherwise, girih seven designs look like girih extended designs, and arabesque. The π/5 girih system and π/7 girih seven system complement each other.

2011 Nobel Prize for Chemistry

The blogger, SIRENDER, sent me this link to the press release on the 2011 Nobel prize for chemistry. It's cool that they mention Islamic mosaics. I don't think they had. It's my understanding that there's really no evidence that Islamic architects and artisans knew the significance of aperiodic tilings. Peter Lu and Paul Steinhardt wrote in 2007 that Darb-i Imam is the only known example of a perfect quasi-crystalline pattern in Islamic art. I wonder if the Alhambra, aperiodic example cited by the press release is a newer discovery. Still, I'm glad the press release from Sweden makes the connection.

Speaking of tilings, here's another girih seven pattern, this time with no gaps in the underlying structure. Since my last post I've added a tile that makes it easier to avoid gaps, though tilings with the new set of heptagon based tiles are still more difficult than pentagonal tilings.

Girih Seven

The images below are my first examples of a girih tile system based on the heptagon. Girih tiles have interior angles that are multiples of π/5. In the examples shown here I've applied the girih concept to tiles with angles that are multiples of π/7. Like girih tiles these have interior decoration (strapwork) that make a pattern without the tile edges. The midpoints of every tile edge have two lines coming from the edge, always at the same angle.

Girih tiles are equilateral, but I extended the tile set to include scaling tiles, allowing me to repeat the tiles at different scales while maintaining an edge-to-edge tiling. This means I can create girih patterns with variation in density of line. Girih seven continues this system. The tile set includes a heptagon, tetradecagon, elongated hexagon, rhombus, and bow tie. I've added a second rhombus, and I'm experimenting with several scaling tiles, including a trapezoid and five-sided kites.

Girih patterns are all straight lines, but starting with girih extended and now with girih seven I've added arabesque versions. Girih patterns are ideally suited for conversion to arabesque using Bézier curves. Since the patterns cross tile boundaries in straight lines, curves are tangent at the boundaries, creating a continuous flow throughout the pattern.

I've found that it's more difficult to create tilings with girih seven. Girih extended made it easy to develop patterns, without gaps. The girih seven example shown below includes a few small gaps, but I think they hardly detract from the finished designs. In at least one case I broke the rule requiring all pattern lines to meet tile edges at the same angle.